Rational points of definable sets and Diophantine problems Jonathan Pila University of Bristol MODNET Barcelona, 4 November 2008
Key points * Upper bounds for the number of rational points of height ≤ T on certain non-algebraic sets X ⊂ R n . * Guiding idea: A“transcendental” set has “few” rational points “in a suitable sense”. * Connection with transcendence theory. * Connection with Manin-Mumford conjecture and other results in diophantine geometry. Plan I Curves II Higher dimensions – Main result III Wilkie’s conjecture IV Manin-Mumford conjecture V Andre-Oort-Manin-Mumford type results i
Notation X ( Q , T ) = { x ∈ X : x ∈ Q n , H ( x ) ≤ T } where X ⊂ R n and height H ( x ) is defined by H ( a 1 /b 1 , . . . , a n /b n ) = max( | a i | , | b i | ) for a i , b i ∈ Z , b i � = 0 , gcd( a i , b i ) = 1 , i = 1 , . . . , n. (Not projective height.) The counting or density function of X , for T ≥ e to avoid trivialities: N ( X, T ) = # X ( Q , T ) . Seek upper bound estimates for N ( X, T ). Constants c ( . . . ) may differ at each occurence. ii
I. Curves Bombieri+JP (1989): results for integer points on the homothetic dilation tX = { ( tx 1 , . . . , tx n ) : ( x 1 , . . . , x n ) ∈ X } . (where t ≥ 1) of a graph X : y = f ( x ) , x ∈ I = [ a, b ] . � tX ∩ Z 2 � , as t → ∞ , for Upper bounds for # * f smooth and convex (won’t discuss) * f real analytic * upper bounds for � X ( Z ) ∩ [0 , T ] 2 � # when f is algebraic (mention briefly) I.1
Transcendental analytic curves Consider X : y = f ( x ) , x ∈ I = [ a, b ] where function f is real-analytic and non-algebraic. Theorem. We have, for every ǫ > 0 , � tX ∩ Z 2 � ≤ c ( f, ǫ ) t ǫ # Note: t ≥ 1 need not be an integer. Theorem. (JP, 1991) For every ǫ > 0 , N ( X, T ) ≤ c ( f, ǫ ) T ǫ f ( x ) = e x then (Hermite-Lindemann) If e.g. the only algebraic point of X is (0 , 1). At other extreme, constructions going back to Weierstrass give: entire transcendental f with f ( Q ) ⊂ Q . (van der Poorten...) Little control of height in such constructions. I.2
Key to method X ( Q , T ) is contained in few (i.e. ≤ c ( X, ǫ ) T ǫ ) intersections of X with plane algebraic curves of suitable degree. Lemma. Let X : y = f ( x ) be C ∞ on [0 , 1] and ǫ > 0 . There is a d = d ( ǫ ) : for every T ≥ 1 , � X ( Q , T ) ⊂ X ∩ V V with the union over O f,ǫ ( T ǫ ) plane algebraic curves V of degree d (possibly reducible). I.3
Proof of Lemma. Consider points P i = ( x i , y i ) ∈ X, i = 1 , . . . , E. They lie on a plane algebraic curve of degree d iff the matrix x 2 y 2 x d y d ( 1 x i y i x i y i . . . . . . i ) , i i i i = 1 , . . . , E , has rank < D = ( d + 1)( d + 2) / 2. If not, have D points with ∆ = det ( φ j ( x i )) � = 0 where the φ j ( x ) are the D functions of the form x µ f ( x ) ν , 0 ≤ µ, ν ≤ d . If P i ∈ X ( Q , T ), the entries in a row have a common denominator ≤ T 2 d . So T 2 dD | ∆ | ≥ 1 . I.4
Mean value statement: if φ j are functions with D − 1 continuous derivatives on an interval containing x i then ∆ 1 0!1! . . . ( D − 1)! det ( φ ( i − 1) V ( x i ) = ( ζ ij )) . j for some suitable intermediate points ζ ij ∈ [0 , 1], V ( x i ) the Vandermonde determinant. In our case: φ j ( x ) = x µ f ( x ) ν , 0 ≤ µ, ν ≤ d . If the x i ∈ I, ℓ ( I ) ≤ r , T − dD ≤ | ∆ | ≤ C ( f, d ) r D ( D − 1) / 2 . I.5
Conclusion: if ℓ ( I ) ≤ C ′ ( f, d ) T − 2 dD/ ( D ( D − 1)) , the points of X ( Q , T ) in I all lie on one curve of degree d . The interval [0 , 1] is covered by C ′′ ( f, d ) T 2 dD/ ( D ( D − 1)) such intervals, and since D = ( d + 1)( d + 2) / 2, the exponent 2 dD D ( D − 1) goes to zero as d → ∞ . Remark. For given ǫ do not need C ∞ : need C D , D <<>> 1 /ǫ 2 , and c ( X, ǫ ) depends on the size of the derivatives of f up to order D − 1. I.6
( Theorem. For ǫ > 0 , X ( Q , T ) ≤ c ( f, ǫ ) T ǫ . ) Proof of Theorem. Choose d = d ( ǫ ) : X ( Q , T ) is contained in c ( f, ǫ ) T ǫ algebraic curves V of degree d . X is transcendental: X ∩ V is finite for any V of degree d . Uniform bound # X ∩ V ≤ C ( d ) for any curve V of degree d by compactness . Then N ( X, T ) ≤ C ( d ) c ( f, ǫ ) T ǫ . I.7
Cannot be much improved. If ǫ ( t ) : [1 , ∞ ) → R is positive, monotonically decreasing to 0, have X : y = f ( x ) , x ∈ [0 , 1] transcendental real-analytic, and a (lacunary) sequence T j such that N ( X, T j ) ≥ T ǫ ( T j ) . j E.g. with ǫ ( t ) = (log t ) − 1 / 2 , gives an example X : y = f ( x ) , x ∈ [0 , 1] satisfying no estimate N ( X, T ) ≤ C (log T ) c . Cf. results of Surroca: better estimates do hold on a sequence of T i → ∞ . I.8
Algebraic curves Theorem. (EB+JP 1989, JP 1996) Suppose f ∈ Z [ x, y ] is absolutely irreducible of degree d , X = { ( x, y ) : f ( x, y ) = 0 } . Then ≤ c ( d ) T 1 /d (log T ) 2 d +3 . � X ( Z ) ∩ [0 , T ] 2 � # Exponent 1 /d is best possible : y = x d . Improvements: JP, Walkowiak (by Heath-Brown method) – application to Hilbert irreducibility. Heath-Brown (2002): a p -adic version of method for rational points on projective varieties in all dimensions, in particular Theorem. (Heath-Brown, 2002) For X ⊂ P 2 irreducible, degree d X ( Q , T ) ≤ c ( d, ǫ ) T 2 /d + ǫ . Exponent 2 /d best possible: y = x d . I.9
Point of estimate: uniformity Siegel/Faltings: finiteness of X ( Z ) (or X ( Q )) for g > 0 but not good uniformity as curve varies with fixed degree. These uniform bounds are crude but useful, especially in higher dimensional problems, e.g. Waring type problems (e.g. Browning, Greaves, Hooley, Skinner-Wooley, Vaughan-Wooley), and Hilbert irreducibility (e.g. work of Schinzel- Zannier, Walkowiak). Heath-Brown’s results have also been useful in further work (HB, Browning, Salberger,...) Breaking 1 /d, 2 /d uniformly when genus g > 0: Helfgott-Venkatesh, Ellenberg-Venkatesh. Bombieri-Zannier: E ( Q ). Schmidt conjecture: c ( d, ǫ ) T ǫ for g > 0. I.10
II. Higher dimensions Seek to generalize N ( X, T ) ≤ c ( X, ǫ ) T ǫ to suitable “transcendental analytic” X ⊂ R n . Consider e.g. surface X ⊂ R 3 X : z = f ( x, y ) , ( x, y ) ∈ [0 , 1] 2 . Straightforward: X ( Q , T ) contained in O X,ǫ ( T ǫ ) intersections of X with algebraic hypersurfaces V of degree d ( ǫ ), where the implied constant depends on sizes of derivatives of f ( x, y ) up to order D <<>> d n . (Determinant ∆, expand entries in Taylor srs.) Repeat the argument for these intersections: semi-analytic curves X ∩ V ? II.1
Leads to: (bounded) semi-analytic sets in R n . Then projections: (bounded) subanalytic sets in R n . These are contained in the globally sub-analytic sets in R n . This class has “good” properties: dimension theory, stratification, cell decomposition, and strong finiteness properties —sets have just finitely many connected components — (get e.g. uniform bounds for intersections with an algebraic curve) Globally subanalytic sets: an example of an o-minimal structure over R . II.2
O-minimal structures over R Definition. A pre-structure is a sequence S = ( S n : n ≥ 1), each S n is a collection of subsets of R n . A pre-structure S is called a structure (over R ) if, for all n, m ≥ 1, (1) S n is a boolean algebra (2) S n contains every semialgebraic subset (3) if A ∈ S n and B ∈ S m , then A × B ∈ S n + m (4) if m ≥ n, A ∈ S m then π ( A ) ∈ S n , where π : R m → R n is projection on first n coords A structure is called o-minimal if (5) The boundary of every set in S 1 is finite. First 4 axioms: S admits various constructions, condition 5 is the “minimality” condition. X ⊂ R n is definable in S if X ∈ S n . II.3
Examples. Semi-algebraic sets: Tarski-Seidenberg R an , the globally subanalytic sets: Gabrielov R exp : the sets definable using y = e x : Wilkie R an , exp : generated by R an and R exp together : van den Dries-Macintyre-Marker Richer examples. No “largest” o-minimal struc- ture: Rolin, Speissegger, Wilkie II.4
Some problems: * Curves X ∩ V are not presented as graphs with uniformly bounded derivatives (indeed they may be singular). (The hypersurfaces V that occur vary with T .) This can be fixed. * Surface X may contain semi-algebraic sets of positive dimension, e.g. lines. These may contain >> T δ rational points up to height T for some δ > 0. This cannot be fixed! II.5
The “algebraic part” Definition. The algebraic part X alg of a set X is the union of all connected semialgebraic subsets of positive dimension . Seek: for suitably “nice” X ⊂ R n , and ǫ > 0, N ( X − X alg , T ) ≤ c ( X, ǫ ) T ǫ . Crude analogue of the special set V special of V in diophantine geometry . V special =Zariski closure of � of images in V of non-constant rational maps of P m , Abelian varieties. Bombieri-Lang Conjecture: ( V − V sp )( Q ) is finite. Curves: Mordell Conjecture (Faltings’s Theorem). In higher dimensions, it is open. “Geometry governs arithmetic” II.6
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